CCSS say the following about what students should be able to do concerning the volume of a cylinder.
8.G.C.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
HS.G-GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
HS.G-GMD.A.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*
The high school standards with an asterisk indicate that the standard is a modeling standard and should be linked to “everyday life, work, and decision-making”.
Our learning targets for the modeling unit are the following:
Level 4: I can use geometry to solve a design problem and make valid conclusions. G-MG 3
Level 3: I can estimate and calculate measures as needed to solve problems. G-MG 2, G-MG 3
Level 2: I can decompose geometric shapes into manageable parts. G-MG 2
Level 1: I can create a visual representation of a design problem. G-MG 1
How did you learn about the volume of a cylinder?
Many students have been given the formula for the volume of a cylinder, V=πr2h and then asked to calculate the volume of cylinders given the length of the radius and height. For example, what is the volume of a cylinder with a radius of 5 in. and a height of 4 in.?
What can learning about the volume of a cylinder look like in a math class using CCSS?
Students made sense of the volume formulas during our Geometric Measure and Dimension unit. For this lesson, we started with a 3-Act lesson by Dan Meyer. You can read more about 3-Acts here if you are interested.
The question about which cylinder holds more popcorn isn’t a new question. I’ve used this question for years in geometry. But how the lesson plays out in class when I’m focused on providing my students an opportunity to model with mathematics and I’m paying close attention to the modeling cycle is different than simply posing the question to my students as I had in the past.
I had recently read a blog post by Michael Pershan where he talks about the difference between asking students “what do you wonder” and “what’s an interesting question we could ask”. I showed my students the first act of the video by Dan Meyer and asked “what question could we explore”. I agree with Michael that changing the wording here a little gets students to think about the math from the beginning.
how much popcorn could fill the cylinder1
Is there a difference in volume 1
How much popcorn went into the cylinders? 1
Do the cylinders hold the same amount of whatever he was pouring in them? 1
Are the areas of the 2 cylinedes the same? 1
how much can the cylinders hold 1
whether the two cylinders hold the same amount of popcorn 1
will they hold the same amount of popcorn 1
WHICH CYLINDER HOLDS THE MOST POPCORN? 1
how much cereal can they hold and is it equal or is one greater 1
are the volumes of the two cylinders the same 1
what is the volume of each cylinder? 1
Do both cylinders have the same volume? 1
do the cylinders have the same volume 1
Is the volume the same for both cylinders? 1
Are the volumes of the cylinders equivalent? 1
Are the volumes of both tubes the same? 1
were the volumes of the 2 cylinders equal? 1
Last year, I simply asked, “What is your question?” While the questions in red certainly aren’t bad questions, they don’t focus on the math that we can explore in the lesson from watching the video. I can see a difference between the prompts.
why is he using popcorn to fill the paper 1
What exactly was the purpose of that? 1
Are the volumes of the cylinders the same? 1
were the amonts of popcorn equal1
what are the heights of the cylinders? 1
What is the volume of the space in the cylinder not filled with popcorn? 1
Are the volumes of these cylinders the same? 1
Will the two cylinders hold the same amount of food? 1
is he pouring popcorn? 1
What was he pouring into the paper cylinders? 1
Which way should you hold the paper in order for it to hold the most popcorn? 1
what are the dimensions of the paper 1
are the volumer of the cylinders the same? 1
is the volume the same 2
Do they have the same volume? 1
Is the video showing us the different volumes 1
Is the volume of the cylinders the same? 1
do cylinders hold the same amount of popcorn? 1
what is he trying to do? 1
whats he trying to do? 1
which cylinder can hold the most popcorn? 1
What is the volume of 1 popcorn kernal. 1
how much popcorn could fit in the bowls 1
Do the cylinders hold the same amount of popcorn? 1
Do the cylinders have the same volume 1
do both cylinders hold the same amount of popcorn things. 1
why is he doing this?
which will hold more?
whats wrong with him? 1
how did the paper not fall apart 1
Do the cylinders have the same volume? 1
what is the radius of each cylinder? 1
So we continued, exploring which container will hold more popcorn. Before we started calculating, students made a guess as to which they thought would hold more popcorn. I sent a Quick Poll asking whether container A would hold more, container B would hold more, or they would both hold equal amounts.
This year’s results:
Last year’s results:
So what would normally happen next is that I would give students measurements so that they could do some calculations for which container holds more popcorn. But instead, I asked students what information they needed to explore the question.
YOU NEED THE AREA OF BASE TIMES HEIGHT OF EACH 1
height hnd radious 1
height and radius 3
radius and height of both 2
what are the dimentions of the paper, the average volume of each piece of popcorn1
both radii, height 1
radii and heights of each 1
length and width of the paper 1
height and radius of both 2
radius and height 5
is it an average size piece of paper? how much do they overlap? 1
area and height 1
the radii and the height 1
diameter of both
height of both 1
radius and height of both cylinders 3
If it is a regular size of paper (8.5×11) 1
radius, height, size of the paper 1
the dimensions of the paper 1
height and width of paper 1
I want their radii and their heights. 1
And then I gave them the information. For this task, I gave everyone the same information, but on some of our modeling tasks, I gave each team only their requested information. (More about that in future posts.)
Container A is made from an 11-in. x 8.5-in. sheet of paper. Container B is made from an 8.5 in. x 11-in. sheet of paper. Students began to calculate with their teams and construct a viable argument as to which container held more popcorn.
They answered a second Quick Poll.
As the teams finished, they started thinking about another question: Can a rectangular piece of paper give you the same amount of popcorn no matter which way you make the cylinder? Prove your answer.
The 3 who still said that the containers would hold equal amounts showed me their work and ultimately corrected their miscalculation.
Students then watched the video of Act 3 where the conflict was resolved.
Teams then decided which of the following questions they wanted to explore next:
- How many different ways could you design a new cylinder to double your popcorn? Which would require the least extra paper?
- Is there a way to get more popcorn using the exact same amount of paper? How can you get the most popcorn using the same amount of paper?
- How many more pieces of popcorn will the first container hold?
For this part, I provided a bag of popped popcorn. By the end of class, we had a whole class discussion on the plan that each team used to answer their chosen question.
What’s most significant about this lesson is not the engaging way that students got to learn but what the students did learn. Several students made comments about this lesson on their end-of-unit survey:
Popcorn Picker was very helpful in helping me learn the targets for this unit. It helped me realize that although two objects may have the same surface area, their volumes may not be the same.
Popcorn Picker definitely helped me understand how exactly different dimensions affect the volume of a cylinder even though the dimensions are nearly the same. Using the piece of paper to compose a cylinder using 8.5 and 11 as two different circumferences as well as the height helped me see that the volumes will be different.
One activity that personally really helped me was whenever we put popcorn into two tubes. One of the was a normal sheet rolled up vertically and the other was horizontally. I learned that even tho they had the same dimensions at first, the one folded horizontally held more pieces of popcorn in the end.
I think the lesson for the popcorn in 11F helped me meet learning targets because it taught me that flipping the dimensions actually changes the volume.
I have learned that two sheets of paper with the same dimensions, but different orientations do not hold the same amount of popcorn.
This student reflection makes me realize how important it is for students to think about what information they need to solve a problem instead of always being given the information from the beginning:
I’ve ;earned how to divide a complex geometric objects into parts and calculate It’s volume. I can find out the necessary information needed to solve this kind of problem and how to use them to solve the problem. I can apply math to every day life and model with mathematics. I can also make visual representation of a design problem.
And so the journey to provide students opportunities to model with mathematics continues, with much gratitude for those who are creating lessons here, here, and here for the rest of us to try with our students …